The bakery with missing buns
Three picnic bags, one bun per bag. Why does Pip’s invoice list only one bun?
The formula in this story
Story 2 of 8
What you will take away
Explain why the multiplier goes to both items inside parentheses.
Read one piece at a time. There is no timer. “Help” stays beside you throughout.
Before reading, how does it feel? (optional)
See the five reading steps
Choose a section directly
What do the words mean?
The distributive law says multiplication spreads over addition. ab means a × b; parentheses group a sum as one input.
What are real numbers?
Real numbers are the numbers on the number line: integers, fractions and numbers such as √2. Our formulas apply to these numbers, even when the cake picture uses only positive lengths.
Want a hand with the prerequisites?The cake scale conspiracy
A suspicious plan
Mika packs three bags, each with two rolls and one bun. Pip: “Three times two, plus one. Seven!”
Open every bag: each contains both b and c.
The clue
The buns protest. There is one bun in every bag, not one for the entire picnic.
Sort by kind without adding or losing anything. There are a copies of each kind.
The twist
Mika counts by bags: 3 × (2 + 1). Pip counts by species: 3 × 2 + 3 × 1. Nine either way. No refunds required.
Count the two groups together: ab + ac. The general real-number law is stated in the proof below.
Change the evidence
Try examples to build intuition. Checking numbers is not the proof for every value.
Step 1 of 3
With a nonnegative integer number of bags, write a copies of (b + c).
The diagram illustrates the displayed argument. Numerical examples do not replace its hypotheses or prove every case.
The proof, with all its steps
Claim and conditions
For real a, b, c, a(b + c) = ab + ac. The bag argument explains the counting case of nonnegative integers.
- With a nonnegative integer number of bags, write a copies of (b + c).
- Regroup all the b terms and all the c terms. This is ab + ac: the same objects counted in two ways.
- For arbitrary real numbers, distribution is a basic law of their arithmetic, not a conclusion from a picture of positive bags. We use that law in the later proofs.
Pip has one more idea…
Which invoice correctly expands 4(x + 2)?
Read the explanation for 4x + 2
The 2 belongs to every one of the four bags. It must also be multiplied by 4.
Read the explanation for 4x + 8
Yes: 4x + 4 × 2. Neither item escapes the four bags.
What makes more sense now?
Two small checks, then your own reflection. You can use the hints and try again. This records these questions, not a score for all of mathematics.
How does it feel now? (optional)
Your record for this story
Your first answers and your latest answers will appear here.
Where does your curiosity go next?
No score, no locked doors. You can return to any definition.
These are educational arguments and numeric experiments, not live Lean verification. The linked knowledge pages distinguish formal source excerpts from explanations.